Cusp types of arithmetic hyperbolic manifolds

We establish necessary and sufficient conditions for determining when a flat manifold can occur as a cusp cross-section within a given commensurability class of arithmetic hyperbolic manifolds of simplest type. This reduces the problem of identifying which commensurability classes of arithmetic hype...

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Bibliographic Details
Published in:arXiv.org 2024-12
Main Authors: McCoy, Duncan, Sell, Connor
Format: Article
Language:English
Subjects:
Arithmetic
Cross-sections
Cusps
Manifolds
Online Access:Get full text
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Summary:We establish necessary and sufficient conditions for determining when a flat manifold can occur as a cusp cross-section within a given commensurability class of arithmetic hyperbolic manifolds of simplest type. This reduces the problem of identifying which commensurability classes of arithmetic hyperbolic manifolds can contain a specific flat manifold as a cusp cross-section to a question involving rational representations of the flat manifold's holonomy group. As applications, we prove that a flat manifold \(M\) with a holonomy group of odd order appears as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds if and only if \(b_1(M)\geq 3\). We also provide examples of flat manifolds that arise as cusp cross-sections in a unique commensurability class of arithmetic hyperbolic manifolds and exhibit examples of pairs of flat manifolds that can never appear as cusp cross-sections within the same commensurability class.
ISSN:2331-8422